High-intensity scaling in UV-modified QED

QED perturbation theory in a background field has been conjectured to break down for sufficiently high field intensity. The high-intensity behavior of a field theory is however intertwined with its high-energy (UV) behavior. Here we show that a UV modification of QED changes the high-intensity behaviour of observables. Specifically we study non-linear Compton scattering in a constant crossed field in QED with an additional Pauli term. In the UV modified theory the cross section exhibits a faster power-law scaling with intensity than in ordinary QED.

The paper is organized as follows. In Sect. II we discuss the physics of the exact solutions to the Dirac-Pauli equation in a background plane wave. In Sect. III we use these solutions to calculate the probability of our chosen test process, nonlinear Compton scattering, specifying to the case of a CCF. We derive the asymptotic scaling of the nonlinear Compton probability in Sect. IV and conclude in Sect. V.

II. QED WITH A PAULI TERM
The Lagrangian of QED with an additional (dimension 5) Pauli term is in which κ is the dimensionless Pauli coupling, σ µν = i 4 [γ µ , γ ν ], and Λ is the cutoff scale as described in the introduction. For convenience we redefine κ → κΛ/m.
We include also a background plane wave field A ext , which shifts A → A + A ext in the interaction terms of (1) [42][43][44][45]. (Later we will specialise to a CCF.) We are interested in the limit in which this field becomes strong, in a sense to be defined, but which amounts to the effective coupling between the background and matter eventually becoming (much) larger than unity. Hence we work in the Furry picture, in which all background-matter couplings are treated exactly, as part of the 'free' theory, while interactions between the quantised fields are treated in perturbation theory as usual. As such, fermion propagators and external legs become 'dressed' to all orders in the background field, in our case through both the QED vertex A µ ext γ µ and through the Pauli vertex σ µν F µν ext .
A. Intensity effects due to Pauli-term In the Furry picture, external fermion legs are given by incoming/outgoing solutions Ψ to the Dirac equation in the chosen background, which now includes the Pauli term. Writing eA µ ext = a µ the equation is with f µν the field strength of a µ . The factor of 1/e appears in the Pauli term just because we have absorbed e into the potential. Our chosen background is a (for the moment arbitrary) plane wave described by a µ ≡ a µ (n · x) in which n 2 = 0, n · a = 0, and a has only spatial components. As is standard, we use light-front coordinates (see [46,47] for reviews) such that x · n = x + = x 0 + x 3 , and the remaining spatial directions are Momenta have components p ± = (p 0 ± p 3 )/2 and p ⊥ = (p 1 , p 2 ). The two nonzero components of a µ are simply the integrals of the two electric field components of the wave, from x + = −∞, see e.g. [48].
To understand the physics introduced by the Pauli term, it is useful to briefly review that of the Volkov wavefunctions, which are the solutions to (2) with the Pauli term switched off. (The high symmetry associated with plane waves guarantees the (super)integrability of both classical and quantum equations of motion [49][50][51].) For an incoming electron of initial momentum p µ the appropriate Volkov solution is [52] The current of these wavefunctions recovers the classical, on-shell, kinematic momentum π µ of an electron in a plane wave background, i.e.
However, while interaction with a plane wave can change electron momentum, it cannot change the quantum spin state of the electron [53]. To see this, first write the spin structure in (3) as It is easily checked that / πu π = mu π , hence u π is a 'free' spinor for the (time-dependent) on-shell momentum π µ and (5) is not a trick of notation. It is convenient to use a lightfront helicity basis [54], in which the spinors are eigenstates of the lightfront helicity operator L p = 2h p · W/m, that is the Pauli-Lubanski vector W µ contracted with h µ p where [54,55], in which p is understood as the momentum of the state acted on. The eigenvalues of L p are ±1, and explicit forms of the corresponding eigenspinors u p± are given in Appendix A. (For discussions of other relativistic spin operators see [54,56].) It is then easy to verify that so that helicity is preserved in the plane wave background. As such the Volkov solutions do not describe helicity-flip transitions, for which either loop corrections or photon emission is needed [53]. We now return to the Dirac equation with Pauli term (2). For a linearly polarised field (as we adopt below) A µ ext = A(x + ) µ with 2 = −1, n · = 0, the solution ψ p (x) to (2) is given by replacing, in (3), u p → U p where [57] It can be checked that the current of these solutions is the same as that of the Volkov solutions. The physical content of the literature result (8) is made transparent using our helicity basis. Using either the explicit representation given in Appendix A, or the defining properties of the states, one can show that Hence the 'cosine' term in (8) preserves the helicity of the state, while the 'sine' term flips it. The extra spin structure due to the Pauli term therefore amounts to a rotation matrix in helicity space: From (5), this result extends immediately to the spinors u π± . Physically, the Pauli term therefore causes an initial helicity eigenstate to evolve into a superposition of helicity states as the particle propagates through the plane wave. In standard QED, such effects arise, implicitly and usually considered only perturbatively, through one-loop corrections, which generate a Pauli term and thus the anomalous magnetic moment of the electron.

III. NONLINEAR COMPTON SCATTERING
In this section we will calculate the probability of nonlinear Compton scattering, that is the emission of a photon from an electron incident on a plane wave, here considered in the long wavelength limit, i.e. as a CCF.
The field is described by the potential a µ = eE µ x + , where, without loss of generality, · x = x 1 . For an electron of momentum p µ , observables in a CCF depend on the 'quantum nonlinearity parameter' χ defined by We focus on the CCF case as there is mounting evidence that universal power-law scaling at large χ is absent for non-constant fields [29,30,[58][59][60]. In a CCF, the electron self-energy loop shows the typical behaviour stated by the Narozhny-Ritus conjecture, which is a large-χ scaling with αχ 2/3 . The same scaling is inherited, via the optical theorem, by nonlinear Compton scattering at tree level. Another reason to focus on CCFs is their common use in laser-plasma simulation codes [61,62] in the form of the locally constant field approximation (LCFA) [63]. Let ψ p± (x) be the helicity eigenstates given by (8) and (3). The nonlinear Compton scattering amplitude is in which µ and ε µ are the momentum and polarisation vector of the outgoing photon, respectively. On the lower line the left hand diagram represents the standard QED vertex −ieγ µ and the right hand diagram the Pauli vertex iκγ µ / /m. We consider the nonlinear Compton probability summed and averaged over spins and polarisations. While the Pauli term simply induces a rotation in helicity space, the spin-summed probability can still exhibit nontrivial effects because the rotation is (light-front) time dependent, and therefore cannot simply be factored out of the amplitude. By standard arguments, see e.g. [64], the probability can be brought to the form (14) in which the momentum p µ of the outgoing electron is eliminated using conservation of momentum in the field, and s := n · /n · p = − /p − is the light-front momentum fraction of the emitted photon. The modifications with respect to standard QED are encoded in the quantity Υ, which contains the (time-dependent) Dirac structure coming from the vertices and the Volkov-Pauli solution (3) and (8): writing π for the classical momentum related to p as π is to p in (4), In contrast to standard QED (κ = 0), the new Dirac structure introduces trigonometric functions of light-front time, seen in (16), into Υ, under the integrals of (14). This will have considerable impact on the evaluation of the emitted photon momentum integrals. Performing the Gaussian integral over ⊥ in (14) eliminates any term linear in ⊥ from Υ and yields a multiplicative factor for any term quadratic in ⊥ . Now, some terms in Υ are proportional to ·π p (x + ), ·π p (y + ) or products thereof, like the exponent of (14). Normally, for κ = 0, these terms are total derivatives when integrating over x + or y + and can be discarded [48,65,66]. However, due to the presence of the trigonometric functions in (16), stemming from the Pauli term, some contributions remain after integration by parts. To illustrate, one such term is proportional to (the unwritten exponent is exactly as in (14)), The following standard change to average and relative phase variables, brings the probability into 'Airy form', in whichΥ is a sum of powers of θ, each carrying a trigonometric factor; for example, the term in (17) contains the factor sin(kθ/ √ z) in which we define Note that the argument of such trigonometric functions depends on both the Pauli coupling and (through z) on χ. If not for the trigonometric factors inΥ, the θ-integrals in the probability could be expressed in terms of Airy functions using the integral representations (with an iε prescription understood) − i n 2π dθ θ −n e iθz+iθ 3 /3 = ∞ z dz 1 · · · ∞ zn−1 dz n Ai(z n ) =: Ai n (z).
However, by writing the trigonometric functions in exponential form, we see that their effect is just that of a finite difference operator acting on the Airy functions, e.g., To make our expressions more managable, we introduce the following shorthand notation for this operator, and others that appear, which ultimately arise from terms containing cos 2∆, sin 2∆, cos 2 ∆, sin 2 ∆, respectively -as such, the operators obey the same relational identities as the trigonometric functions themselves. The final result for the probability has three parts, proportional to e 2 , eκ and κ 2 , respectively, corresponding to the three terms in |M| 2 , Explicitly, we find The usual CCF result is recovered from (29) by setting κ = 0, upon which C → 1 and D − → 0. The shifts of k/ √ z in the arguments of the Airy functions encode a change in the photon spectrum, relative to ordinary QED, through the s-dependence in z. Further, one sees in the same combination k/ √ z ∼ κχ 1/3 /e an explicit interplay between the highenergy modification of the theory and the intensity dependence. The high precision to which the anomalous magnetic moment of the electron is known would, phenomenologically, limit κ (equivalently k) to small values. This means that there will be, typically, only minor phenomenological effects from the additional terms in (29)-(31). However, our interest is in the regime of extreme field strengths, and we will now show that the high-intensity behaviour of the probability above is dominated by terms coming from the Pauli interaction. Viewed another way, we will show that strong fields essentially enhance κ.

IV. HIGH-INTENSITY BEHAVIOUR
In this section we establish the asymptotic strong field behaviour of the emission probability (27). To extract this from (29)-(31) we must first understand integrals of the type where A is Ai or one of its derivatives or integrals, and λ is either ±k or 0; the latter is needed because even some of the κ-dependent terms of (29)-(31) have zero shift of the Airy function. After a change of variables s → t, where the integrals (32) take the form, where now z = t 2/3 and all χ-dependence has been moved outside A. We first consider λ = 0. In order to obtain the asymptotic behaviour of our integrals we will need to distinguish between 'small' and 'large' t; thus we introduce a cutoff L, below which A(z) can be approximated by a Taylor-MacLaurin series, its order N determined by the desired accuracy of approximation. This divides the integral as For χ large enough, the 1 in the denominator is negligible in the second term (t > L), so that J 1 is O(χ −a ). By simply performing the integral in the first term (change variables to t = 1 + χt and use the binomial series), one finds Since the scaling in χ decreases with β, thus with b, the dominant contribution to J 0 comes from the leading term of the Taylor-MacLaurin series, and the scaling of J can now be read off from (36). We now turn to the case λ = 0. Here we can divide the integration as the error in the argument of A being less than ε in the first term (t < ε). By the same reasoning as before, J 1 = O(χ −a ). If λ > 0, an asymptotic expansion for A around +∞ can be used, viz., While the integration extends down to 0, if χt 1 does not hold, then the log of the integrand is O(− √ χ), so non-negligible contributions come only from t 1/χ, hence J 0 is again O(χ −a ). If λ < 0 and A is Ai or one of its derivatives, the asymptotic expansion around −∞ is The integral over u := t −1/2 can be estimated by writing the integrand in exponential form and closing the contour through a quarter circle in the first or fourth quadrant to guarantee convergence, see Fig. 1. As b > −1, the integral along the arc does not contribute in the asymptotic limit. (The integrand has poles at ±i √ χ, but these are outside the contour.) (In any case, the logs of the residues go like − √ χ, and so would yield exponentially suppressed contributions at large χ.). Now, on the vertical segments of the contour, the integrand goes like e −|u| , so contributions outside u χ are exponentially suppressed. Thus, we again find J 0 = O(χ −a ). On the other hand, if A is Ai 1 , the asymptotic expansion is, for C a constant, 40) and the scaling of the first term can be read off from (36). Finally, the Airy differential equation implies that Ai 2 (x) = −x Ai 1 (x) − Ai (x), reducing terms with Ai 2 to previously treated cases.
With the preceding results, we can go through each term in the three contributions (29)- (31) to the probability, and identify the dominant terms at large χ. These are: • At order e 2 : both Ai terms scale like χ 2/3 . Viz., according to (36) and likewise, • At order eκ: the dominant term comes from, assuming k > 0, where for small t, using the expansion of Ai 1 around −∞, ε 0 dt after performing the integral with the constant term. (The cos term is O(χ 0 ), as explained above.) We note that this is a faster scaling than in standard QED.
• At order κ 2 : the dominant term arises from To show this, observe that for t above some cutoff and χ large enough, χt 1, thus the tail of the integral goes like χ 0 . Below the cutoff, expanding Ai(z) as in (35) and performing the integral shows that it is O(χ 4/3 ), as in (36). For sufficiently large χ, this will become the dominant term of the whole probability.
Numerical data which confirm the above are presented in Fig. 2 -the dominant scaling, for large χ, is ∝ χ 4/3 and comes from the Pauli vertex in the amplitude (mod-squared).
We note that the scaling of the majority of terms without shift operators is the same as would be obtained from naive power counting (with z ∼ χ −2/3 ). There are however, from (36), terms which scale logarithmically, exemplified in Fig. 3, whereas a naive estimate would suggest O(χ 0 ). Further, for terms with shift operators, power counting is often incorrect. An example of this is illustrated in Fig. 2(e) where the λ = 0 scaling is χ 4/3 , but the λ = 0 scaling is χ 0 . The reason naive power counting fails for λ = 0 is that it replaces the Airy function with its value at 0 when χt 1, but with λ > 0 this is never the case for large χ, and with λ < 0 the integrand is, rather, rapidly oscillating around 0, unless it is the asymptotically constant Ai 1 . The subtleties of the λ → 0 limit can be illustrated by plotting the argument z + λ/ √ z of A as a function of z, Fig. 5 in Appendix B; for any non-zero λ, this function bends away from z, such that the limit λ → 0 is qualitatively different from any non-zero λ.
By performing the integral (43), we can work out the coefficient of the dominant (χ 4/3 scaling) term in the total probability of nonlinear Compton scattering. This gives us the large-χ dominant rate R (where P = R dϕ) for nonlinear Compton scattering with a Pauli term, (c) f = sχz 1/2 Ai 2 (z + λ/ √ z) (e) f = χsz −1/2 Ai(z + λ/ √ z)  as can be confirmed from the numerical data in Fig. 2(e), and which can be compared to the usual CCF result [10,63,67], If we simply equate αχ 2/3 = κ 2 χ 4/3 , we see that the dominance of the Pauli term sets in for χ values of order (e/κ) 3 , or, reinstating the original 'new-physics' scale Λ from (1), χ ∼ (e/κ) 3 (Λ/m) 3 . The numerical results also show that subdominant terms, due to the Pauli coupling, remain of magnitude comparable to leading terms for fairly large χ, depending on κ; see Appendix B for details. This reflects the interplay between energy and intensity: extremely high intensity is required for the high-energy effects to dominate.

V. CONCLUSIONS
We have shown that changing the high-energy behaviour of QED induces a change in its high-intensity behaviour. Due to the presence of the introduced Pauli term, observables such as scattering rates scale differently with intensity compared to ordinary QED.
Considering nonlinear Compton scattering at tree level in Furry picture perturbation theory, we have established the leading-order intensity dependence of the scattering probability (14) at high intensity. Regarding the three contributions (29)-(31) and identifying their dominant high-χ scaling in (41)- (43), we see that each Pauli vertex introduces an additional factor χ 1/3 to the high intensity scaling: the dominant terms coming from the QED vertex squared, the interference term, and the Pauli vertex squared, cf. (28), scale as eχ 1/3 2 , κχ 2/3 eχ 1/3 , and κχ 2/3 2 respectively. Now, there are two ways of viewing high-energy corrections to nonlinear Compton scattering: either through the explicit Pauli term; or through loop effects which, recall, induce a vertex with the same form, ψσF ψ. Diagrammatically, we may therefore represent the high-energy corrections as: Mod-squaring the bottom line yields the three terms of (46). In the upper line, the QED one-loop vertex correction in a CCF scales like e 3 χ 2/3 at high χ [68], which reproduces the scaling of the Pauli vertex upon replacing the QED vertex count, e 3 , by the Pauli coupling κ. Thus, (46) provides a useful consistency check: both the QED and effective field theory view lead to the same high-χ behavior. Of course, the effective field theory can only be expected to capture part of the full theory; for instance, we should not expect that the spectrum in (27) agrees in detail with that computed using the loop correction. As seen in Fig. 2, the high-intensity power law scaling only sets in for very large values of χ, say for χ ∼ 10 3 . At such extreme intensities, backreaction on the strong field is likely to be non-negligible [69][70][71], and a treatment beyond the external field approximation [45] is necessary. To date the impact of back-reaction does not seem to have been considered in the context of the Narozhny-Ritus conjecture, and it would be interesting to address this. (For a toy model illustrating that backreaction cannot be neglected see [72].) One could also consider, with the inclusion of a Pauli term, higher orders in perturbation theory, or resummation to all orders as in [11]. Given our results, it would also be interesting to revisit the case of non-constant fields, where the asymptotics depend on energy and intensity separately [29,58]; this clearly ties into the interrelation of high intensity and high energy we have studied here. For any λ = 0, the argument of A(z + λ/ √ z) bends away from the straight line for small z, making the λ = 0 case qualitatively different from the λ = 0 case in integrals like (32).